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...ysis has increasingly shown that the traffic streams in modern broad-band networks exhibit long-tailed (subexponential) characteristics. For the case of Ethernet traffic such results were examined in =-=[LTW93]-=-. These statistical results have stimulated research in queueing analysis under the heavy-tailed (non Cramér) assumptions. Queueing analysis with self-similar long-range dependent arrival processes ap...

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...given by F(x) = 1− l(x) α ≥ 0, xα where l(x) : R+ → R+ is a function of slow variation, i.e., limx→∞l(δx)/l(x) = 1,δ > 1. These functions were invented by Karamata [KAR30] (the main reference book is =-=[BGT87]-=-). TheotherexamplesincludelognormalandsomeWeibulldistributions(see[KLU88,JLA95g]). 3. Analysis of the Aggregate Arrival Process This section consists of two parts. The first part is contained in Secti...

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..., beginning of the last activity period until time zero, i.e., Dc(0) = ∫ 0 t b 0 where tb 0 represents the beginning of the activity period that is still active at t = 0. Now, by Theorem 4.3, pp. 64, =-=[ASM87]-=-, it follows that the process {Tn + τon n ,−∞ < n < ∞} is also a stationary Poisson process with the same rate Λ. Therefore, the process A ∞,s t is reversible. This implies that Dc(0) is equal in dist...

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...stributions In what follows we will state a few important results from the literature on subexponential distributions. The general relation between S and L is the following. Lemma 3 (Athreya and Ney, =-=[ATN72]-=-) S ⊂ L. Lemma 4 If F ∈ L then (1−F(x))e αx → ∞ as x → ∞, for all α > 0. Note: Lemma 4 clearly shows that for long-tailed distributions Cramér type conditions are not satisfied. The proof of the follo...

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...ocess. Hence, investigating computationally tractable exact and approximate solution techniques are needed. For Markovian (fluid) On-Off processes a thorough investigation of this problem was done in =-=[AMS82]-=-. Many other results for multiplexing Markovian OnOff processes followed. These led to the Equivalent Bandwidth theory for Markovian, or, in general, exponentially bounded arrival processes; extensive...

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...er r, any M/G/∞ process is uniquely defined with Possion arrival times {Tn}, and the lengths of On periods {τon n }; likewise, the pair {Tn}, {τon n } uniquely defines a compound Poisson process (see =-=[CIN75]-=-, pp. 90), that is a piecewise constant process with Poisson jump times {Tn}, and jump sizes {τon n }. Hence, proving (4.22) is equivalent to 18 Z d = Z y +Z y , (4.23) where Z,Z y ,Z y , are the comp...

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... be a sequence of i.i.d. random variables that are driving a queueing process (Lindley’s recursion) Qn+1 = (Qn +Xn) + , n ≥ 0, (4.1) where q + = max(0,q). According to the classical result of Loynes’ =-=[LOY62]-=- under the stability condition EXn < 0 this recursion admits a unique stationary solution, and for all initial conditions P[Qn ≤ x] converges to the stationary distribution P[Q ≤ x]. For the rest of t...

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...e obtained up to now apply for IR. In addition, directly from the definition it can be shown that F ∈ IR, ∫ ∞ ¯F(t)dt < ∞, ⇒ F1 ∈ IR. 0 Under the general large deviation Gärtner-Ellis conditions (see =-=[WES95]-=-) on the arrival process, it can be proved that the queue length distribution is exponentially bounded. To avoid statingGärtner-Ellisconditions, wewill define anarrival processet tobeexponential, if w...

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... + = max(q,0). From the recursion above, it is clear that the process Qt is essentially the same as the G/G/1 workload process. Hence, by the fundamental stability theorem of Loynes (see Chapter 2 in =-=[BAB94]-=-) there exists a unique stationary process {Qs t ,−∞ < t < ∞} (P[Qs t ≤ x] = P[Q ≤ x]) that satisfies (4.4) (or equivalently (4.2)). In the rest of thepaper, whenever we refer to Qt, we will actually...

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...ractable approach using fluid renewal type models in which renewal times are long-tailed has been explored in [ANA95, HRS96]. Queueing results in these two papers rely onthe classical result by Pakes =-=[PAK75]-=- onthe subexponential (long-tailed) asymptotics of the waiting time distribution in a GI/GI/1 queue or in earlier work of Cohen [COH73] which considered a regularly varying GI/GI/1 queue. TheresultofP...

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...dual process may be truly innocuous (like an On-Off process). The complex autocorrelation structure of the aggregate process obtained by multiplexing long-tailed On-Off processes has been examined in =-=[HRS96]-=-. General bounds for multiplexing long-tailed fluid processes have been derived in [CHW95]. In [BOX96] a limiting case of an infinite number of OnOff processes with regularly varying On distribution h...

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...x→∞ ∗2 (x) = 2, (2.2) 1−F(x) where F∗2 denotes the 2-nd convolution of F with itself, i.e., F∗2 (x) = ∫ [0,∞) F(x−y)F(dy). The class of subexponential distributions was first introduced by Chistyakov =-=[CHI64]-=-. The definition is motivated by the simplification of the asymptotic analysis of convolution tails. One of the best known examples of distribution functions in S (and L) are functions of Regular Vari...

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...results in these two papers rely onthe classical result by Pakes [PAK75] onthe subexponential (long-tailed) asymptotics of the waiting time distribution in a GI/GI/1 queue or in earlier work of Cohen =-=[COH73]-=- which considered a regularly varying GI/GI/1 queue. TheresultofPakeshasbeengeneralizedtoaMarkovmodulatedsetting[AHK94,JLA95g]. In [AHK94] the subexponential asymptotics of a Markov modulated M/G/1 qu...

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...varying GI/GI/1 queue. TheresultofPakeshasbeengeneralizedtoaMarkovmodulatedsetting[AHK94,JLA95g]. In [AHK94] the subexponential asymptotics of a Markov modulated M/G/1 queue was investigated. Work in =-=[JLA95g]-=- further generalized these results to Markov modulated G/G/1 queues. In the same paper it was shown that a subexponential GI/GI/1 queue is invariantunder Markov modulation. In other words, a subexpon...

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... theorem relates the tail behavior of a distribution function to the asymptotic behavior of its Laplace transform at the origin. For convenience we state the following result due to Bingham and Doney =-=[BID74]-=- ([BGT87], pp. 333). Let F beadistribution functionon[0,∞), andlet ˜ F(s)beitsLaplace-Stieltjes transform. Denotebymn = EX n = ∫ [0,∞) xn dF(x),n = 0,1,.... Whenmn < ∞, ˜ F(s)maybeexpanded26 in a Tay...

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...ar Pareto family); F ∈ R−α if it is given by F(x) = 1− l(x) α ≥ 0, xα where l(x) : R+ → R+ is a function of slow variation, i.e., limx→∞l(δx)/l(x) = 1,δ > 1. These functions were invented by Karamata =-=[KAR30]-=- (the main reference book is [BGT87]). TheotherexamplesincludelognormalandsomeWeibulldistributions(see[KLU88,JLA95g]). 3. Analysis of the Aggregate Arrival Process This section consists of two parts. ...

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...λEI N,on ˜ FN,1(s) . ˜FN,1(s) = π N 0 ˜ Ftr,N(s) 1−(1−π N 0 ) ˜ Ftr,N(s) ,7 or, in the time domain ¯FN,1(t) = π N 0 ∞∑ n=1 (1−π N 0 )n−1 F ∗n tr,N (t). (3.8) Now, F ∈ Sd implies F1 ∈ S (Theorem 1.1, =-=[KLU89]-=-), which by (3.7), and Lemma 5 (ii) (A), yields Ftr,N ∈ S. Hence, by Theorem 14 (A) and (3.8) it follows that ¯FN,1(t) ∼ π −N 0 ¯Ftr,N(t) ∼ Nπ1 1−π N 0 ¯F1(t) as t → ∞, (3.9) where the second asymptot...

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...all α > 0. Note: Lemma 4 clearly shows that for long-tailed distributions Cramér type conditions are not satisfied. The proof of the following result can be found in Embrechts, Goldie and Veraverbeke =-=[EGV79]-=-. Lemma 5 Let F ∈ S. Then, (i) If G is a probability distribution such that ¯ G(x) = o( ¯ F(x)) as x → ∞, then F ∗G(x) ∼ ¯F(x) as x → ∞. (ii) If limx→∞ ¯ G(x)/ ¯ F(x) = c ∈ (0,∞), where G is a distrib...

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...ss with On arrival rate r. In this subsection we assume that Off periods are also general (not necessarily exponential). (A general storage model in a two state random environment was investigated in =-=[KEW92]-=-.) Then, if we observe the queue at the beginning of On periods, the queue length Q P n evolves as follows (P stands for Palm probability [BAB94]). 13 Q P n+1 = (QP n +(r −c)τon n −cτoff n ) + , n ≥ 0...

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...ocesses (sources); the main QOS parameter (performance measure) for this queueing system is the buffer occupancy probability distribution. The problem of multiplexing dates back to [RUB73, COH74]. In =-=[COH74]-=-, Cohen obtained a complete Laplace transform solution to this problem! (More recently, he revisited this problem in [COH94].) However, inverting the Laplace transform is usually a very tedious proces...

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...e distribution in a GI/GI/1 queue or in earlier work of Cohen [COH73] which considered a regularly varying GI/GI/1 queue. TheresultofPakeshasbeengeneralizedtoaMarkovmodulatedsetting[AHK94,JLA95g]. In =-=[AHK94]-=- the subexponential asymptotics of a Markov modulated M/G/1 queue was investigated. Work in [JLA95g] further generalized these results to Markov modulated G/G/1 queues. In the same paper it was shown ...

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...structure of the aggregate process obtained by multiplexing long-tailed On-Off processes has been examined in [HRS96]. General bounds for multiplexing long-tailed fluid processes have been derived in =-=[CHW95]-=-. In [BOX96] a limiting case of an infinite number of OnOff processes with regularly varying On distribution has been investigated. In the same paper (see also [BOX97]) a case of two processes, one of...

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... the aggregate process obtained by multiplexing long-tailed On-Off processes has been examined in [HRS96]. General bounds for multiplexing long-tailed fluid processes have been derived in [CHW95]. In =-=[BOX96]-=- a limiting case of an infinite number of OnOff processes with regularly varying On distribution has been investigated. In the same paper (see also [BOX97]) a case of two processes, one of which had r...

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...bution function F is intermediate regular varying F ∈ IR if 15 limliminf δ↓1 t→∞ ¯F(δt) ¯F(t) = 1. Remark: For recent results on distributions of intermediate regular variation we refer the reader to =-=[CLI94]-=-. Some basic properties of IR are: IR ⊂ S; R ⊂ IR. Also, it is not very difficult to see that IR ⊂ Sd. Therefore, all of the results that we have obtained up to now apply for IR. In addition, directly...

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...d processes have been derived in [CHW95]. In [BOX96] a limiting case of an infinite number of OnOff processes with regularly varying On distribution has been investigated. In the same paper (see also =-=[BOX97]-=-) a case of two processes, one of which had regularly varying On periods and the other had exponential On periods, has been solved. Similar scenario with intermediately regularly varying On periods ha...

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... distribution. The problem of multiplexing dates back to [RUB73, COH74]. In [COH74], Cohen obtained a complete Laplace transform solution to this problem! (More recently, he revisited this problem in =-=[COH94]-=-.) However, inverting the Laplace transform is usually a very tedious process. Hence, investigating computationally tractable exact and approximate solution techniques are needed. For Markovian (fluid...

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...cesses, one of which had regularly varying On periods and the other had exponential On periods, has been solved. Similar scenario with intermediately regularly varying On periods has been examined in =-=[RSS97]-=-. The literature does not explicitly give precise asymptotic results for the case of multiplexing two or more long-tailed processes. From a mathematical point of view the new results in this paper wer...

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...r of customers in service in an M/G/∞ queue in which the customer service requirement has the same distribution as τon and the arrival rate is Λ. (For recent asymptotic results on M/G/∞ processes see =-=[PAM97]-=-.) Note that Theorem 3 implies that lim N→∞ lim t→∞ λN→Λ P[I N,on > t] P[τ on > t] = eΛτon. (3.14) However, this does not necessarily imply that we can interchange the limit and derive the asymptotics...

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...bexponential distributions (and further references) can be found in [CLI86, KLU88]. For a recent survey of the application of subexponential distributions in queueing theory the reader is referred to =-=[ASM96]-=-. B: Proof of Theorem 6 As we have already mentioned the proof of this result is based on Karamata’s Tauberian/Abelian theorem for distribution functions of regular variation. This theorem relates the...

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...e analysis of a subexponential semi-Markov fluid queue ([JLA95g]). Further generalizations of the result in [JLA95g] to arrival processes with a more complex dependency structure were investigated in =-=[ASC97]-=-. Asymptotic expansion refinements of Pakes’ result can be found in [WIT92, ACW94]. The analysis of a fluid queue in which more than one long-tailed process is multiplexed appearsto beavery difficult ...